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Metamath Proof Explorer

Theorem ltrniotaval

Description: Value of the unique translation specified by a value. (Contributed by NM, 21-Feb-2014)

Ref Expression
Hypotheses ltrniotaval.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrniotaval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrniotaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrniotaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
ltrniotaval.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
Assertion ltrniotaval ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) = 𝑄 )

Proof

Step Hyp Ref Expression
1 ltrniotaval.l ⊢ ≤ = ( le ‘ 𝐾 )
2 ltrniotaval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 ltrniotaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrniotaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 ltrniotaval.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
6 1 2 3 4 cdleme ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
7 nfriota1 ⊢ Ⅎ 𝑓 ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
8 5 7 nfcxfr ⊢ Ⅎ 𝑓 𝐹
9 nfcv ⊢ Ⅎ 𝑓 𝑃
10 8 9 nffv ⊢ Ⅎ 𝑓 ( 𝐹 ‘ 𝑃 )
11 10 nfeq1 ⊢ Ⅎ 𝑓 ( 𝐹 ‘ 𝑃 ) = 𝑄
12 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
13 12 eqeq1d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑃 ) = 𝑄 ↔ ( 𝐹 ‘ 𝑃 ) = 𝑄 ) )
14 11 5 13 riotaprop ⊢ ( ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 → ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) = 𝑄 ) )
15 14 simprd ⊢ ( ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 → ( 𝐹 ‘ 𝑃 ) = 𝑄 )
16 6 15 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) = 𝑄 )